Solve 5x^2+50x+100=2500000

$$5 { x }^{ 2 } +50x+100=2500000$$

Solve for x (complex solution)

$$x=\sqrt{500005}-5\approx 702.110316712$$

$$x=-\left(\sqrt{500005}+5\right)\approx -712.110316712$$

Steps Using the Quadratic Formula

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$5x^{2}+50x+100=2500000$$

Subtract 2500000 from both sides of the equation.

$$5x^{2}+50x+100-2500000=2500000-2500000$$

Subtracting 2500000 from itself leaves 0.

$$5x^{2}+50x+100-2500000=0$$

Subtract 2500000 from 100.

$$5x^{2}+50x-2499900=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, 50 for b, and -2499900 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-50±\sqrt{50^{2}-4\times 5\left(-2499900\right)}}{2\times 5}$$

Square 50.

$$x=\frac{-50±\sqrt{2500-4\times 5\left(-2499900\right)}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-50±\sqrt{2500-20\left(-2499900\right)}}{2\times 5}$$

Multiply -20 times -2499900.

$$x=\frac{-50±\sqrt{2500+49998000}}{2\times 5}$$

Add 2500 to 49998000.

$$x=\frac{-50±\sqrt{50000500}}{2\times 5}$$

Take the square root of 50000500.

$$x=\frac{-50±10\sqrt{500005}}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{-50±10\sqrt{500005}}{10}$$

Now solve the equation x=\frac{-50±10\sqrt{500005}}{10} when ± is plus. Add -50 to 10\sqrt{500005}.

$$x=\frac{10\sqrt{500005}-50}{10}$$

Divide -50+10\sqrt{500005} by 10.

$$x=\sqrt{500005}-5$$

Now solve the equation x=\frac{-50±10\sqrt{500005}}{10} when ± is minus. Subtract 10\sqrt{500005} from -50.

$$x=\frac{-10\sqrt{500005}-50}{10}$$

Divide -50-10\sqrt{500005} by 10.

$$x=-\sqrt{500005}-5$$

The equation is now solved.

$$x=\sqrt{500005}-5 x=-\sqrt{500005}-5$$

Steps for Completing the Square

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$5x^{2}+50x+100=2500000$$

Subtract 100 from both sides of the equation.

$$5x^{2}+50x+100-100=2500000-100$$

Subtracting 100 from itself leaves 0.

$$5x^{2}+50x=2500000-100$$

Subtract 100 from 2500000.

$$5x^{2}+50x=2499900$$

Divide both sides by 5.

$$\frac{5x^{2}+50x}{5}=\frac{2499900}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$x^{2}+\frac{50}{5}x=\frac{2499900}{5}$$

Divide 50 by 5.

$$x^{2}+10x=\frac{2499900}{5}$$

Divide 2499900 by 5.

$$x^{2}+10x=499980$$

Divide 10, the coefficient of the x term, by 2 to get 5. Then add the square of 5 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+10x+5^{2}=499980+5^{2}$$

Square 5.

$$x^{2}+10x+25=499980+25$$

Add 499980 to 25.

$$x^{2}+10x+25=500005$$

Factor x^{2}+10x+25. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+5\right)^{2}=500005$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+5\right)^{2}}=\sqrt{500005}$$

Simplify.

$$x+5=\sqrt{500005} x+5=-\sqrt{500005}$$

Subtract 5 from both sides of the equation.

$$x=\sqrt{500005}-5 x=-\sqrt{500005}-5$$

Solve for x

$$x=\sqrt{500005}-5\approx 702.110316712$$

$$x=-\sqrt{500005}-5\approx -712.110316712$$

Steps Using the Quadratic Formula

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$5x^{2}+50x+100=2500000$$

Subtract 2500000 from both sides of the equation.

$$5x^{2}+50x+100-2500000=2500000-2500000$$

Subtracting 2500000 from itself leaves 0.

$$5x^{2}+50x+100-2500000=0$$

Subtract 2500000 from 100.

$$5x^{2}+50x-2499900=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, 50 for b, and -2499900 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-50±\sqrt{50^{2}-4\times 5\left(-2499900\right)}}{2\times 5}$$

Square 50.

$$x=\frac{-50±\sqrt{2500-4\times 5\left(-2499900\right)}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-50±\sqrt{2500-20\left(-2499900\right)}}{2\times 5}$$

Multiply -20 times -2499900.

$$x=\frac{-50±\sqrt{2500+49998000}}{2\times 5}$$

Add 2500 to 49998000.

$$x=\frac{-50±\sqrt{50000500}}{2\times 5}$$

Take the square root of 50000500.

$$x=\frac{-50±10\sqrt{500005}}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{-50±10\sqrt{500005}}{10}$$

Now solve the equation x=\frac{-50±10\sqrt{500005}}{10} when ± is plus. Add -50 to 10\sqrt{500005}.

$$x=\frac{10\sqrt{500005}-50}{10}$$

Divide -50+10\sqrt{500005} by 10.

$$x=\sqrt{500005}-5$$

Now solve the equation x=\frac{-50±10\sqrt{500005}}{10} when ± is minus. Subtract 10\sqrt{500005} from -50.

$$x=\frac{-10\sqrt{500005}-50}{10}$$

Divide -50-10\sqrt{500005} by 10.

$$x=-\sqrt{500005}-5$$

The equation is now solved.

$$x=\sqrt{500005}-5 x=-\sqrt{500005}-5$$

Steps for Completing the Square

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$5x^{2}+50x+100=2500000$$

Subtract 100 from both sides of the equation.

$$5x^{2}+50x+100-100=2500000-100$$

Subtracting 100 from itself leaves 0.

$$5x^{2}+50x=2500000-100$$

Subtract 100 from 2500000.

$$5x^{2}+50x=2499900$$

Divide both sides by 5.

$$\frac{5x^{2}+50x}{5}=\frac{2499900}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$x^{2}+\frac{50}{5}x=\frac{2499900}{5}$$

Divide 50 by 5.

$$x^{2}+10x=\frac{2499900}{5}$$

Divide 2499900 by 5.

$$x^{2}+10x=499980$$

Divide 10, the coefficient of the x term, by 2 to get 5. Then add the square of 5 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+10x+5^{2}=499980+5^{2}$$

Square 5.

$$x^{2}+10x+25=499980+25$$

Add 499980 to 25.

$$x^{2}+10x+25=500005$$

Factor x^{2}+10x+25. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+5\right)^{2}=500005$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+5\right)^{2}}=\sqrt{500005}$$

Simplify.

$$x+5=\sqrt{500005} x+5=-\sqrt{500005}$$

Subtract 5 from both sides of the equation.

$$x=\sqrt{500005}-5 x=-\sqrt{500005}-5$$